Question

A cistern has two inlets A and B which can fill it in 12 minutes and 15 minutes respectively. An outlet C can empty the full cistern in 10 minutes. If all the three pipes are opened together in the empty tank, how much time will they take to fill the tank completely?

Answer 1

$$\begin{gathered} \mathrm{Inlet}\mathrm{A}\mathrm{can}\mathrm{fill}\mathrm{the}\mathrm{cistern}\mathrm{in}12\mathrm{minutes}. \\ \mathrm{Inlet}\mathrm{B}\mathrm{can}\mathrm{fill}\mathrm{the}\mathrm{cistern}\mathrm{in}15\mathrm{minutes}. \\ \mathrm{Outlet}\mathrm{C}\mathrm{empties}\mathrm{the}\mathrm{filled}\mathrm{cistern}\mathrm{in}10\mathrm{minutes}. \\ \text{Part of the cistern filled by inlet A in one minute}=\frac{1}{12} \\ \text{Part of the cistern filled by inlet B in one minute}=\frac{1}{15} \\ \text{Part of the cistern emptied by outlet C in one minute}=-\frac{1}{10}(\mathrm{water}\mathrm{flows}\mathrm{out}\mathrm{from}\mathrm{C}\mathrm{and}\mathrm{empties}\mathrm{the}\mathrm{cistern}) \\ \text{Part of the cistern filled in one minute with A, B and C working together}=\frac{1}{12}+\frac{1}{15}-\frac{1}{10}=\frac{5+4-6}{60}=\frac{3}{60}=\frac{1}{20} \\ \mathrm{The}\mathrm{time}\mathrm{required}\mathrm{to}\mathrm{fill}\mathrm{the}\mathrm{cistern}\mathrm{with}\mathrm{all}\mathrm{inlets},A,\mathrm{B}\mathrm{and}\mathrm{C},\mathrm{open}\mathrm{is}20\mathrm{minutes}. \end{gathered}$$